Algebraic Structure of Holomorphic Poisson Cohomology on Nilmanifolds

It is proved that on nilmanifolds with abelian complex structure, there exists a canonically constructed non-trivial holomorphic Poisson structure. We identify the necessary and sufficient condition for its associated cohomology to be isomorphic to the cohomology associated to trivial (zero) holomorphic Poisson structure. We also identify a sufficient condition for this isomorphism to be at the level of Gerstenhaber algebras.


Introduction
The study of deformation of real Poisson bracket and its related cohomology theory was initiated by Lichnerowicz et. al. long ago [7] [19]. Deformation of Poisson brackets in complex algebraic category emerged 20 years ago [22]. There has been associated deformation theory within the category of complex analytic geometry [1] [15] [25].
In the past fifteen years, holomorphic Poisson structure is also conceived as an object in the realm of generalized geometry [12] [13] [14] . In this vein, one studies deformation within the framework of Lie bi-algebroids [20]. There has been work along this line by various authors [9] [10] [11] [16] [17] [18] [24]. In this note, the authors continue to treat holomorphic Poisson structures within the realm of generalized complex structures, and extend the work in [24].
A holomorphic bi-vector field Λ is Poisson if its self bracket is equal to zero: [[Λ, Λ]] = 0. When Λ = 0, it is a trivial case and we get nothing but a complex manifolds. As the condition [[Λ, Λ]] = 0 is homogeneous, when Λ = 0, for any complex number t, tΛ is a holomorphic Poisson structure. When t = 0, they are all equivalent and could be considered as a deformation of the trivial holomorphic Poisson structure given by t = 0. Associated to each non-trivial holomorphic Poisson structure, one has a holomorphic version of Lichnerowicz's differentials ∂ tΛ [3] [18] [19]. They act on the space of sections of exterior bundles of (1, 0)-vectors and (0, 1)-forms.
for any non-negative integer k. They form an elliptic complex, and yield cohomology spaces. ∂ tΛ is a perturbation of the classical operator ∂. When t = 0, the resulting cohomology spaces are H k 0 = ⊕ p+q=k H q (M, Θ p ) where Θ p is the sheaf of germs of sections of the p-th exterior power of the holomorphic tangent bundle of the underlying complex manifold M. For each t = 0, we have cohomology spaces H k tΛ . They are all isomorphic to H k Λ . Similar to the classical Lichnerowicz differential, ∂ Λ = ∂ + ad Λ where ad Λ is the Schouten bracket of the Poisson structure Λ with tensorial objects. In the holomorphic setting, the cohomology H k Λ is the hyper-cohomology of a spectral sequence of a bi-complex. ad Λ is also the operator on the first page of the spectral sequence: The general goal of this piece of work is to understand the relation between the algebraic structures on H k Λ and those on H k 0 . Our focus is on nilmanifolds with abelian complex structures. As found in [4] [5] [26] and explained in Section 3 in this paper, on such a manifold there exists a global frame of (1, 0)-forms {ω 1 , . . . , ω n } such that for each k, dω k is type-(1, 1) and it is in the exterior algebra generated by {ω 1 , . . . , ω k−1 , ω 1 , . . . , ω k−1 }. We call such a basis an ascending basis. Let {V 1 , . . . , V n } be the dual basis. Its complex linear span is denoted by g 1,0 . As to be explained in Definition 1 below, we find that Λ = V n ∧ V n−1 is a canonically defined holomorphic Poisson structure.
Our first main result is to identify a necessary and sufficient condition for the map d p,q 1 of the spectral sequence associated to this particular holomorphic Poisson structure to be identically zero.
Theorem Let M be a nilmanifold with abelian complex structure. Let Λ be the canonical holomorphic Poisson structure associated to an ascending basis of the complex structure. The map d p,q 1 is identically zero for all p, q if and only if there exists a vector X in g 1,0 such that ad Λ ω n = ∂X. Whenever the vector X exists, we call it a potential vector.
The above theorem appears as Theorem 3 in Section 4. Additional effort is needed to prove that the degeneracy of the spectral sequence actually lead to a vector space isomorphism as stated in Theorem 4. The proof of these two results occupies much of Section 4.
They generalize a result in [24].
As explained in Proposition 1, given Λ = V n ∧ V n−1 , a characterization of the potential vector X is the set of equations below.
This characterization also corrects an error in Proposition 9 of [24].
After the conditions on a vector space isomorphism is clarified, we complete an analysis in Theorem 5 on whether the isomorphism φ in (1) is an isormophism of algebras. We state the result below.
Theorem Suppose that Λ is the canonical holomorphic Poisson structure associated to an ascending basis of an abelian complex structure on a nilmanifold. If X is a potential vector, then the map φ is an isomorphism of Gerstenhaber algebras when The above result upgrades our past observation regarding isomorphism on the level of vector spaces. As an application of the above results, we prove Theorem 6 in Section 6.1 as stated below.
Theorem On a two-step nilmanifold with abelian complex structure, if dω n is a nondegenerate map then the canonical Poisson cohomology with respect to an ascending basis is isomorphic to the one associated to the trivial (zero) Poisson structure as Gerstenhaber algebras.
The above result recovers an ad hoc computation by the first author in his paper [23] on Kodaira surfaces, and give the computation a theoretical understanding.
It also extends and explains various examples of two-steps nilmanifolds found in [24]. At the end of this article we provide abundant examples beyond two-step nilmanifolds on which the conditions for Theorem 5 are satisfied. We also found examples for which the conclusion for Theorem 3 does not hold. Together, they present a rich subject for further investigation.

Algebraic Structures on Poisson Cohomology
Let M be a manifold with an integrable complex structure J. The complexified tangent bundle of M splits into direct sum of the bundle of (1, 0)-vectors T M 1,0 and the bundle of (0, 1)-vectors T M 0,1 . We denote their dual bundles by T M * (1,0) and T M * (0,1) respectively. Their p-th exterior bundles are denoted by T M p,0 , T M 0,p , T M * (p,0) and T M * (0,p) respectively. Define L = T M 1,0 ⊕ T M * (0,1) . Via a natural pairing between forms and vectors, its complex conjugation L is complex linearly isomorphic to its dual L * . The pair L and L forms a complex Lie bi-algebroid with respect to the Courant bracket [8] [20]. It follows the theory of Lie bi-algebroid [20] [21] and there exists a natural differential When the Courant bracket is restricted to L, it is the Schouten bracket [21], also known as Frölicher-Nijenhuis bracket in classical complex deformation theory. It is extended to a differential of exterior algebras: The differential ∂ has the properties such that when it is restricted to (0, 1)-forms, it is the classical ∂-operator in complex manifold theory; meaning that it is the (0, 2)-component of the exterior differential [11]. Similarly, when the Lie algebroid differential is restricted to (1, 0)-vector fields, it is the Cauchy-Riemann operator as seen in [8].
Since ∂ • ∂ = 0, one obtains the Dolbeault cohomology with coefficients in holomorphic polyvector fields. Denoting the sheaf of germs of sections of the p th exterior power of the holomorphic tangent bundle by Θ p , we have In subsequent computations, when p = 0, Θ p represents the structure sheaf O of the complex manifold M.
Due to the compatibility between ∂ and the Schouten bracket [[−, −]], and the compatibility between ∂ and the exterior product ∧ as noted above, the Schouten bracket and exterior product descend to the cohomology space H • (M, Θ • ). In other words, the triple forms a Gerstenhaber algebra.
Suppose that Λ is a holomorphic Poisson bivector field, then [[Λ, Λ]] = 0 and ∂Λ = 0. Consider the map satisfies the condition ∂ Λ • ∂ Λ = 0. It generates a complex, and hence a cohomology space. We denote it by H • Λ (M), and call it the (associated) Poisson cohomology. Furthermore, as Λ is a degree-2 element in the differential Gerstenhaber algebra , ∧), the operator ad Λ satisfies the identities: , ∧) is a differential Gerstenhaber algebra, , ∧) inherits the structure of a Gerstenhaber algebra. Observe that Since ∂ • ∂ = 0, ad Λ • ∂ + ∂ • ad Λ = 0, and ad Λ • ad Λ = 0, the cohomology H • Λ (M) is given by the hypercohomology of the bi-complex C ∞ (M, T M •,0 ⊗ T M * (0,•) ). The first page of related spectral sequence is given by the classical Dolbeault cohomology of (0, q)-forms with coefficients in the sheaf of germs of holomorphic p-vector fields [27]. For in the first page of the spectral sequence is the map ad Λ , which descends from (11) to cohomology level. 3

Abelian Complex Structure on Nilmanifolds
After reviewing some fundamental observations on complex structures on nilmanifolds as given in [26], we set up several technical observations to facilitate the computation in subsequent sections.
A nilmanifold M is a co-compact quotient of a simply connected nilpotent Lie group G. Denote the Lie algebra of the group G by g. We assume that dim R g = 2n.
Denote the dual space by g * . One has the Chevalley-Eilenberg differential d. If α is in g * , X, Y are in g, then by definition It is extended to the Chevelley-Eilenberg complex, The descending central series of g is the chain of ideals defined inductively by g 0 = g and g j = [g j−1 , g] for all j ≥ 1. By definition, the Lie algebra g is s-step nilpotent if g s = {0} and g s−1 = {0}. In particular, the chain of ideals terminates, A real linear map J : g → g with the properties that J • J = −identity and for all is an abelian complex structure. Denote the spaces of (1, 0)-vectors and (0, 1)-vectors respectively by g 1,0 and g 0,1 . The identity (16) above implies that g 1,0 is an abelian complex algebra. Therefore, the only Lie bracket among elements of g C is of the and its complex conjugate, where X ∈ g 1,0 and Y ∈ g 0,1 .
Taking the adjoint of J, one obtains a map, also denoted by J, from g * to itself.
Their p-th exterior products are respectively denoted by g p,0 , g 0,p , g * (p,0) and g * (0,p) . In particular, by taking invariant polyvector fields and invariant forms, we have an inclusion map Theorem 1 [2] The inclusion map ι is a quasi-isomorphism. In other words, it induces an isomorphism of cohomology The Chevelley-Eilenberg complex is complex linearly extended. When the complex structure is abelian as defined in (16), then dα is a type-(1,1) form when α is a type-(1,0) form. In the rest of this article, the following observation will play a crucial role.
In other words, the structure constants A j kℓ are such that for each j, Taking complex conjugation and rearranging indices, we get In particular, As noted in [26], one obtains the basis {ω 1 , . . . , ω n } inductively starting with dω 1 = 0. Therefore, we address this basis for g 1,0 as an ascending basis. Apparently, an ascending basis is not necessarily unique. Let {V 1 , . . . , V n } be the dual basis for g 1,0 . We will refer to it as the (dual) ascending basis for g 1,0 . We will also use the following notations, Their conjugated counterparts t 0,1 , c 0,1 , t * (0,1) and c * (0,1) are naturally defined. With these notations, we consider the contraction of dω n and dω n as linear maps.
Moreover, with the given ordered bases, the matrix representation A n of dω n is A n kℓ acting on row vectors from the right. i.e. if V = c 1 V 1 + · · · + c n−1 V n−1 , then For instance, The matrix representation of dω n is −A T .
From the structure equations (18) and (19), one finds the expression of structure equations in the basis {V 1 , . . . , V n } ∪ {V 1 , . . . , V n } for the complexified Lie algebra It is now apparent that V n is in the center of g C . Moreover, for each ω k in the ascending basis and any Y in g 0,1 , It follows that As a result of (24), Corollary 1 Λ = V n ∧ V n−1 is a holomorphic Poisson bivector.
Proof: Λ is Poisson because the complex algebra g 1,0 is abelian. It is holomorphic because Definition 1 Given an ascending basis {V 1 , . . . , V n−1 , V n } for g 1,0 on a nilmanifold with abelian complex structure, we call Λ = V n ∧ V n−1 its canonical holomorphic Poisson structure, or simply a canonical Poisson structure of the abelian complex structure.

Cohomology of Canonical Poisson Structures
Since the canonical Poisson structure is invariant, given the quasi-isomorphism in Theorem 1, one could reduce the spectral sequence computation with E 1 -terms given in (13) to a computation of invariant objects. Namely, the d 1 map is reduced to The goal of this section is to prove the following theorem, which generalizes a key result in [24].
Theorem 3 Let M be a nilmanifold with abelian complex structure. Let Λ be the canonical holomorphic Poisson structure associated to an ascending basis of the complex structure. The map d p,q 1 is identically zero for all p, q if and only if there exists a vector X in g 1,0 such that ad Λ ω n = ∂X.
We call such X, if exists, a potential vector for the canonical holomorphic Poisson structure. The proof below is similar to, but simplifies an argument in [24] due to the adoption of an ascending basis.
Proof: One direction of the computation is straightforward. Assume that the map d p,q 1 is identically zero for all p, q. When (p, q) = (0, 1), the space H 1 (g 0,0 ) is precisely the space of invariant closed (0, 1)-forms. Given that the complex structure is abelian, it is apparent that every element in an ascending basis {ω 1 , . . . , ω n } is ∂-closed. Furthermore, given that it is an ascending basis, ι Vn dω j = 0 for all 1 ≤ j ≤ n, and ι V n−1 dω k = 0 for all 1 ≤ k ≤ n − 1. (29) Therefore, the only possible non-trivial term in the computation of d 0,1 1 is contributed by It is equal to zero in H 1 (g 1,0 ) as a cohomology class if and only if there exists an element X in g 1,0 such that ad Λ (ω n ) = ∂X.
On the other hand, since Γ p,q is ∂-closed and all (0, 1)-forms of an ascending basis are also ∂-closed, In view of the nature of an ascending basis, every (∂Υ b ) ∧ Ψ b is an element in In particular, It follows from the previous paragraph that In terms of spectral sequence, d p,q 1 (Γ p,q ) = ad Λ (Γ p,q ) ≡ 0 as a cohomology class in H q+1 (g p,0 ), concluding the proof of Theorem 3.
As noted in [24], the bi-grading yields a natural filtration of the cohomology H n Λ : such that There is a natural injective map from ψ : Recall that ∂ Λ = ad Λ + ∂, when we group ∂ Λ Γ by bi-degrees, we find Therefore, ∂Γ m,k−m = 0, and hence Γ m,k−m represents a class in H k−m (g m,0 ). One could verify that it represents a well defined image of ψ(Γ).
To prove that the map ψ is surjective, recall the decomposition of Γ p,q as in (32), or equivalently (34) when q ≥ 1.
Note that α p,q−1 is in g p,0 ⊗ t * (0,q−1) and the complex structure is abelian. The only non-trivial outcome in is contributed by [[V n , ω k ]] and [[V n−1 , ω k ]] for 1 ≤ k ≤ n − 1 in a fixed ascending basis. However, by nature of an ascending basis (20), these items are equal to zero.
It follows that for all q ≥ 1.
It shows that if Γ p,q = ω n ∧ α p,q−1 + β p,q , represents a class in H q (g p,0 ), then is ∂ Λ -closed. It represents a class in such that ψ(Γ) = Γ p,q . Therefore the map ψ is an isomorphism. The map φ is its inverse.
Proposition 1 A vector X is a potential vector of the canonical Poisson structure V n ∧ V n−1 associated to an ascending basis {ω 1 , . . . , ω n } if and only if Proof: Identity (26) states that and {V 1 , . . . , V n } is a basis for g 1,0 , the conclusion of the proposition follows.
Remark The above observation corrects an error in the statement of Proposition 9.2 in [24] for omitting a requirement on the step of the nilmanifold.

Identification of Gerstenhaber Algebras
Let Γ = (Γ p+q,0 , Γ p+q−1,1 , . . . , Γ p+1,q−1 , Γ p,q , . . . , Γ 1,p+q−1 , Γ 0,p+q ) represents an element in ⊕ p+q=k H q (g p,0 ), the proof of Proposition 4 shows that it is lifted to Since Γ p+q,0 is in g p+q,0 and the complex structure is abelian, it is ∂ Λ -closed so long as it is ∂-closed. With Identity (38) above, we conclude that the map φ takes Γ p+q into H p+q Λ . If one arranges φ(Γ p+q ) in terms of the bi-degree (p, q) in the decomposition of Suppose that ad Λ (ω n ) is ∂-exact and X is a potential vector, so that φ is an isomorphism of vector spaces from ⊕ p+q=k H q (g p,0 ) to H k Λ . Recall that ⊕ p+q=k H q (g p,0 ) inherits the structure of a Gerstenhaber algebra. In fact, one may consider it to be the Gerstenhaber algebra of a trivial (zero) holomorphic Poisson structure; H k Λ has its own Gerstenhaber algebra structure. In the next theorem, we identify a situation when φ is an isomorphism of Gerstenhaber algebras.
Theorem 5 Suppose that Λ is the canonical holomorphic Poisson structure associated to an ascending basis of an abelian complex structure on a nilmanifold. If ad Λ (ω n ) = ∂X for a vector X, then the map φ is an isomorphism of Gerstenhaber algebras if the action of ad X = [[X, −]] on t * (0,1) is identically zero.
It follows (39) that On the other hand, (39) and (45) together implies that and It follows that Making use of (46) and (47) and after a tedious computation, we verify that Since the complex structure is abelian, the action of ad X = [[X, −]] on t p,0 is identically zero for all p. Therefore, if the action of ad X on t * (0,1) is also identically zero, then ad X is identically zero on t p,0 ⊗ t * (0,q) for all p, q ≥ 0. It follows that 6 Applications 6.1 Two-steps case As an application of various observations in the previous sections, we focus on twosteps nilmanifold with abelian complex structure. Suppose that the map is non-degenerate, so that dω n is non-degenerate as well. It follows that −ι V n−1 dω n is a non-zero vector. As dω n is non-degenerate, there exists X in t 1,0 such that By nature of an ascending basis and the assumption that the algebra is two-step, dω k = 0 for 1 ≤ k ≤ n − 1. It follows that all conditions in Proposition 1 are satisfied. Therefore, X is a potential vector.
Furthermore, as dω k = 0 for all 1 ≤ k ≤ n − 1 as well, it follows that [[X, ω k ]] = ι X dω k = 0 for all ω k with 1 ≤ k ≤ n − 1. As they span t * (0,1) , then the potential vector satisfies the conditions in Theorem 5. As a consequence, we obtain the following observation.
Theorem 6 On a two-step nilmanifold with abelian complex structure, if dω n is non-degenerate, the canonical Poisson cohomology with respect to an ascending basis is isomorphic as Gerstenhaber algebras to the one associated to the trivial (zero) Poisson structure. i.e.

Four-dimension example
The only non-trivial real four-dimension example is the Kodaira surface, whose corresponding Lie algebra is a two-step nilpotent algebra [26]. In terms of an ascending basis {ω 1 , ω 2 }, its structure equations are where C is a non-zero constant. When {V 1 , V 2 } is the dual basis, it is apparent that Then X = − C C V 2 is a potential vector. In this case, the isomorphism is at the level of Gerstenhaber algebras as predicted in Theorem 6 above, which provides a theoretical explanation for a computation in [23].

A counterexample
Here we present an example to illustrate that the issue in this paper is non-trivial.
Consider a real vector space W 6 spanned by {X 1 , X 2 , X 3 , X 4 , Z 1 , Z 2 }. Define a Lie bracket by Consider J • J = −identity and It is an abelian complex structure. A basis for g 1,0 is The corresponding complex structure equations become Let {ω 1 , ω 2 , ω 3 } be the dual basis for g * (1,0) , then dω 1 = 0, dω 2 = 0, and dω 3 = − 1 2 Since The equation does not have a solution. Therefore, the canonical holomorphic Poisson structure associated to nilmanifold with the given abelian complex structure fails to have a decomposition as noted in Theorem 3.
Suppose that X = c 1 V 1 + c 2 V 2 is a potential vector field of the canonical Poisson structure, it satisfies the following set of constraints.
It terms of coordinates, these constraints become Therefore, c 1 = 0 and c 2 = B/C if BC = 0. In other words, if the map dω 3 from t 1,0 to t * (0,1) is non-degenerate, then the canonical Poisson structure has a potential vector. Moreover, the Hodge-type decomposition is also an isomorphism of Gerstenhaber algebras.
There are two cases when dω 3 degenerates, but is non-trivial.
If B = 0, ad V 2 ω 3 = 0. It follows that ∂ Λ = ∂ + ad Λ = ∂ when it acts on ⊕ p,q B p,q . Therefore, ⊕ p+q=k H q (g p,0 ) is naturally identical to H k Λ . If C = 0 and B = 0, the system (52) does not have a solution.
This set of examples extends our scope beyond two-step nilmanifolds as seen in Section 6.3 above.
This is a two-step nilmanifold. Note that If X = c 1 V 1 + c 2 V 2 is a potential vector field, then the sole constraint is ι X dω 3 = −ι V 2 dω 3 , i.e.
Therefore, if dω 3 is non-degenerate, the canonical Poisson structure has a potential vector.
If dω 3 is a rank-1 map, there are various cases. Instead of investigating every case, we consider a particular situation. If C = 0 and D = 0, the constraints on a potential vector are reduced to • If A = 0 and B = 0, then there is no solution. This case corresponds exactly to the example found in Section 6.3.
The above observation points out the complication of the existence of potential vector when dω n degenerates. It will be a subject for future analysis.